Numerical analysis of vorticity transport and energy dissipation of inner-blade vortex in Francis turbine

The inner-blade vortex results in flow instability when Francis turbines operates at off-design conditions. The numerical method was conducted versus Francis turbine to study the inner-blade vortex characteristics. Firstly, the potential rothalpy gradient (PRG) is applied to analyze the inner-blade vortex evolution innovatively. The PRG is the main power which drives the inner-blade vortex develop in radial direction. The difference of potential rothalpy between hub and shroud curve occurs at 1/3 distance from blade inlet to outlet for deep load condition and 1/4 distance for part load condition. Then, the inner-blade vortex evolution is deeply analyzed based on the vorticity transport equation in cylindrical coordinate system. It indicates that the vorticity transport in radial and axial direction was mainly affected by stretching term and the Coriolis force term makes major contribution to vorticity in circumferential direction. Finally, the energy loss was calculated in Francis turbine by entropy production theory. The relative error was 3.51% and 3.07% for guide vane opening 8.28 degrees and 10.85 degrees between entropy production method and pressure drop method for efficiency calculation. And the interaction characteristics between inner-blade vortex and the energy dissipation was revealed by vorticity transport equation and entropy production method.